On a Linear Differential Equation of the Advanced Type *

The purpose of this short paper is to study a simple differential equation of the advantage type, see Eq. (1). Such equations appear in several branches of applied mathematics, for example, see [ 1 ] for an application in probability. In this paper and for the simple type of equation considered, we show existence of a solution on any interval [0, T], where T is any positive finite constant, and provide and justify the procedure for generating all solutions on [0, T]. There are several papers in the literature dealing with differential equations of the advanced type. A local existence theorem for equations more general that the one considered here is given in Theorem 2 of [2], by using the Schauder fixed point theorem. This result of [2] generalizes a result of [3]. The existence theorem of [3] is a local existence theorem proved under a hypothesis which does not hold in our case (see Remark 1). In [4] an advanced type of differential equation is studied and several asymptotic results are obtained. The analysis of [4] does not apply to the case considered here (see Remark 1). In [5], analytic solutions are considered and it is shown that for the advanced case, they almost never exist. Finally, [6] and [7] consider analytic solutions on the half axis.